Igusa showed in [1]} that the closure of the locus of hyperelliptic Jacobians in
\({\mathcal {A}}_{3}\)  is the zero divisor of a Siegel modular cusp form \(\chi _{18}\)  of degree 3
and weight 18. One can view \(\chi _{18}\)  as a section of the line bundle
\({\det }(\mathbb {E})^{\otimes 18}\)  with \({\mathbb {E}}\)  the rank 3
Hodge bundle on \(\overline{\mathcal {M}}_{3}\) .
It allows us to view \({\mathcal {M}}_{3}\)  as a double cover of \({\mathcal {A}}_{3}\)  obtained by taking a square
root \(\chi _9\)  of \(\chi _{18}\) . Ichikawa showed in 1995 that \(\chi _9\)  is a Teichmüller
modular form of weight 9,
that is, a section of \({\det }({\mathbb {E}})^{\otimes 9}\) 
on \(\overline{\mathcal {M}}_{3}\) , [2]}. An algebraic way to construct it
is by observing that there is a natural morphism of locally free
sheaves of rank 6
\({\rm Sym}^2({\mathbb {E}}) {} \over \longrightarrow  \pi _*(\omega _{\mathcal {C}/{\mathcal {M}}_{3}}^{\otimes 2})\) 
